The standard treatment for optimally debulked Stage III ovarian or primary peritoneal carcinoma is chemotherapy with intravenous Cisplatin and Paclitaxel. The purpose of this study is to determine the effectiveness of administration of chemotherapy into the abdominal cavity in addition to intravenous chemotherapy compared to the current standard intravenous chemotherapy. Because ovarian cancer spreads in the abdominal cavity, giving chemotherapy by infusion into the abdominal cavity may result in a greater dose of the drugs reaching tumor cells. This study will evaluate the patient's response, and survival/disease free interval when treated with standard intravenous Cisplatin and Paclitaxel compared to intravenous Paclitaxel and intraperitoneal Cisplatin and Paclitaxel. The study will also evaluate the impact of treatment on the patient's quality of life.